Is it possible to have a compact Kähler manifold that admits a homogeneous action of a non-reductive Lie group? It seems not to be the case, but a precise argument of reference would be great!
Edit: To clarify, I am asking about smooth actions.
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3$\begingroup$ I guess that "homogeneous" means "transitive". Also you certainly want a faithful action since otherwise there are trivial examples (e.g., any non-reductive Lie group acting on a point). $\endgroup$– YCorCommented Apr 12 at 16:56
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3$\begingroup$ Is your action by smooth isometries, just smooth, or something else? $\endgroup$– LSpiceCommented Apr 12 at 19:41
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2$\begingroup$ If the action is isometric, the isometry group being compact, every transitive subgroup will be reductive. But indeed one should clarify. Does it mean a holomorphic action, regardless of the Kähler structure? $\endgroup$– YCorCommented Apr 12 at 19:43
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